Perturbative S-matrix for massive scalar fields in global de Sitter space

Abstract: We construct a perturbative S-matrix for interacting massive scalar fields in global de Sitter space. Our S-matrix is formulated in terms of asymptotic particle states in the far past and future, taking appropriate care for light fields whose wavefunctions decay only very slowly near the de Sitter conformal boundaries. An alternative formulation expresses this S-matrix in terms of residues of poles in analytically-continued Euclidean correlators (computed in perturbation theory), making it clear that the standard Minkowski-space result is obtained in the flat-space limit. Our S-matrix transforms properly under CPT, is invariant under the de Sitter isometries and perturbative field redefinitions, and is unitary. This unitarity implies a de Sitter version of the optical theorem. We explicitly verify these properties to second order in the coupling for a general cubic interaction, including both tree- and loop-level contributions. Contrary to other statements in the literature, we find that a particle of any positive mass may decay at tree level to any number of particles, each of arbitrary positive masses. In particular, even very light fields (in the complementary series of de Sitter representations) are not protected from tree-level decays.

Overview of the Paper "Perturbative S-matrix for Massive Scalar Fields in Global de Sitter Space"

The paper "Perturbative S-matrix for Massive Scalar Fields in Global de Sitter Space" addresses the construction of a perturbative S-matrix for interacting massive scalar fields within the context of global de Sitter (dS) space. The authors present a formulation of the S-matrix using asymptotic particle states that are defined in the far past and future. This work takes into account the complexities introduced by light fields whose wavefunctions decay slowly near the de Sitter conformal boundaries. The study also suggests an alternative expression for the S-matrix based on residues of poles from analytically-continued Euclidean correlators, aligning the result with the S-matrix in Minkowski space under the flat-space limit.

Key outcomes of the research involve establishing that the S-matrix conforms to CPT symmetry, maintains dS isometries, undergoes proper behavior under perturbative field redefinitions, and retains unitarity. Importantly, this unitarity supports a dS version of the optical theorem. The authors verify these characteristics up to second order in the coupling for a general cubic interaction, considering both tree- and loop-level contributions. Contrary to some claims in existing literature, the authors argue that particles with any positive mass can decay at the tree level into particles of arbitrary positive masses, highlighting that even light fields in the complementary series of dS representations are susceptible to tree-level decay.

Methodology

1. Theoretical Formulation: The paper constructs an S-matrix within the global de Sitter space by defining asymptotic states and applying perturbation theory. A significant part of this involves careful integration and analysis of Klein-Gordon mode functions, ensuring the correct extraction of physical poles signifying stable particles.

2. Handling Asymptotics and IR Divergences: One challenge the authors tackle is the slow decay of wavefunctions at the dS boundaries, which can lead to infrared (IR) divergences. They deploy a projection operator technique to effectively isolate contributions from relevant states while circumventing unphysical divergences associated with complementary series fields.

3. Numerical and Analytical Techniques: Throughout the analysis, the authors employ advanced analytical techniques like Mellin-Barnes representations of hypergeometric functions to evaluate integration contours and residues. This approach allows them to correctly identify singularity structures crucial for defining the asymptotic behavior of the fields.

Results

• General Properties: The S-matrix was shown to possess necessary theoretical properties, such as unitarity, which is confirmed by demonstrating the dS version of the optical theorem through explicit calculations.
• Asymptotic State Construction: The authors provide a comprehensive treatment for constructing asymptotic states such that the S-matrix retains essential field-theoretic properties. Specifically, the requirement for asymptotic states to transform under the de Sitter group was rigorously followed.
• Decay Channels: An insightful conclusion of the study is that light fields are not necessarily protected from decay processes, contrary to some earlier suggestions. The paper argues that decay is possible even for very light fields, and this has implications for understanding particle interactions in curved spacetime scenarios.

Implications and Future Directions

The implications of this research are twofold: firstly, it enriches the conceptual understanding of quantum field theory in curved spacetime, particularly within the framework of expanding cosmological models like de Sitter space. Secondly, it provides a foundational tool that can aid in further theoretical investigations in scenarios where dS space serves as a backdrop, such as studies relating to the early universe, inflationary cosmology, and potentially even quantum gravity theories derived from string theory.

For future studies, the paper opens several avenues: extending the S-matrix framework to incorporate gauge fields or dynamical gravity fields, exploring connections with holographic theories like dS/CFT correspondence, and investigating the non-perturbative definition of the S-matrix, particularly in theories with inherently unstable or exceptional particle spectra.

In conclusion, the paper contributes a significant theoretical advance in the study of quantum fields on de Sitter space, providing a robust mathematical framework for analyzing particle interactions in this geometrically complex setting.